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A277360 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).

%I A277360 #12 Mar 29 2017 09:28:42
%S A277360 1,9,491,64159,15314361,5799651689,3193954129651,2410542221526399,
%T A277360 2388182999073694001,3006071549433968619529,4685653563347872021885371,
%U A277360 8859314350383162594502273439,19975392290718104323103596377961,52949467092712165429316121638458089
%N A277360 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).
%H A277360 Alois P. Heinz, <a href="/A277360/b277360.txt">Table of n, a(n) for n = 0..200</a>
%F A277360 a(n) = (16*n^2-4*n-1)*a(n-1) - n*(4*n-6)*a(n-2) for n>1, a(0)=1, a(1)=9.
%F A277360 a(n) = (2n)! * [x^(2n)] exp(-x/2)/(1-2*x)^(5/4).
%F A277360 a(n) = A277358(2*n).
%F A277360 a(n) ~ sqrt(Pi) * 2^(4*n + 13/4) * n^(2*n + 3/4) / (Gamma(1/4) * exp(2*n + 1/4)). - _Vaclav Kotesovec_, Oct 13 2016
%p A277360 a:= proc(n) option remember; `if`(n<2, 8*n+1,
%p A277360       (16*n^2-4*n-1)*a(n-1)-n*(4*n-6)*a(n-2))
%p A277360     end:
%p A277360 seq(a(n), n=0..15);
%t A277360 a[n_] := a[n] = If[n<2, 8n+1, (16n^2 - 4n - 1) a[n-1] - n (4n-6) a[n-2]];
%t A277360 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 29 2017, translated from Maple *)
%Y A277360 Cf. A277358, A277359.
%K A277360 nonn,walk
%O A277360 0,2
%A A277360 _Alois P. Heinz_, Oct 10 2016